Nonlinear statistical models, Chap. 6, Multivariate nonlinear regression.

(1)

by

A. Ronald Gallant

CHAPTER 6. Multivariate Nonlinear Regression

(2)

This copy is reproduced by special permission of the author and pUblisher from:

A. Ronald Gallant. Nonlinear Statistical Models. New York: John Viley and Sons, Inc, forthcoming.

and is being circulated privately for discussion purposes only. Reproduction from this copy is expressly prohibited.

Please send comments and report errors to the following address: A. Ronald Gallant

Institute of Statistics

North Carolina State University Post Office Box 8203

(3)

Table of Contents

1. Univariate Nonlinear Regression 1.0 Preface

1.1 Introduction

1.2 Taylor's Theorem and Matters of Notation 1.3 Statistical Properties of Least Squares

Estimators

1.4 Methods of Computing Least Squares Estimators 1.5 Hypothesis Testing

1.6 Confidence Intervals 1.1 References

1.8 Indu

2. Univariate Nonlinear Regression: Special Situations 3. A Unified Asymptotic Theory of Nonlinear Statistical

Models Ant icipated Completion Date Completed December 1985 Completed 3 .0 3. 1 3.2 3.3 3.4 3.5 3.6 3.1 3.8 3.9 3.10 Preface Introduction

The Data Generating Model and Limits of Cesaro Sums

Least Mean Distance Estimators Method of Moments Estimators Tests of Hypotheses

Alternative Representations of a Hypothesis Random Regressors

Constrained Estimation References

Index

4. Univariate Nonlinear Regression: Asymptotic Theory 4.0 Preface

4.1 IntrOduction

4.2 Regularity Conditions

4.3 Characterizations of Least Squares Estimators and Test Statistics

4.4 References 4.5 Index

5. Multivariate Linear Models: Review

Completed

(4)

6. Multivariate Nonlinear Models 6.0 Preface

6.1 Introduction

6.2 Least Squares Estimators and Matters of Notation 6.3 Asymptotic Theory

6.4 Hypothesis Testing 6.5 Confidence Intervals

6.6 Maximum Likelihood Estimators

6.7 An Illustration of the Bias in Inference Caused by Misspecification

6.8 References 6.9 Index

7. Linear Simultaneous Equations Models: Review 8. Nonlinear Simultaneous Equations Models

Anticipated Completion Date

Completed

(5)

Chapter

6. Multivariate Nonlinear Regression

All that separates multivariate regression from univariate regression

is a linear transformation.

Accordingly, the main thrust of this chapter is

to identify the transformation, to estimate it, and then to

app~

the ideas

of Chapter 1.

In Chapter 1 we saw that there is little difference between

linear and nonlinear least squares save for some extra

tedi~~

in the

co~pu

(6)

1. INTRODUCTION

In Chapter

1

we considered univariate nonlinear model

t = 1 , 2 , ...

, n .

Here we consider the case where there are

M

such regressions

t = 1,2, ... ,

n;

a

= 1,2, ... , M

that are related in one of two ways.

The first arises most naturally when

Yat

a

=

1, 2, ... , M

represent repeated measures on the same subject, height and weight

measure-ments on the same individual for instance.

In this case one would expect

the observations with the same t index to be correlated, viz

One often refers to this situation as contemporaneous correlation.

The

second way these regressions can be related is through shared parameters.

stacking the parameter vectors and writing

81

82 8=

8 _M

one can have

8 =

g{p)

(7)

with improved efficiency can be obtained; improved in the sense of better

efficiency than that which obtains by applying the methods of Chapter 1 M

times

(Problem 12, Section 3). An example that exhibits these characteristics that we shall use heavily for illustration is the following.

EXAMPLE 1.

(Consumer Demand)

The data shown in Tables la and lb is to

be transformed as follows

y

=

l,n (peak expenditure share) - l,n (base expenditure share)

1 y

=

l,n (intermediate expenditure share) - l,n (base expenditure share)

2 xl

=

l,n (peak price/expenditure)

x

=

l,n (intermediate price/expenditure)

2 x

=

l,n (base price/expenditure)

.

2 As notation, set

Y

=

(~~)

x

=

(:~)

Y

t

=r

l t

)

Clt)

t

1, 2, ... , 224

xt

=

x

2t

=

Y2t

x

3t

These data are presumed to follow the model

Ylt

=

_l,n[(al

+

x'b(1))/(a

3

+

X'b

U ))]

+

elt

Y2t

=

1,r{

_(a2

+

X'b(2))/(a

3

+

x'b(3))]

+

e2t

where

a

=

(:~)

B

ell

b

12 b

13

=

b

23 b

21 b

22 b

(8)

and b(i) denotes the ith row of B, viz.

The errors

are assumed to be independently and identically distributed each with mean

zero and variance-covariance matrix

~

.

There are various hypotheses that one might impose on the model.

Two

are of the nature of maintained hypotheses that follow directly from the

theory of demand and ought to be satisfied.

These are:

Hl : a 3 and b(3) are the same in both equations, as the

notation suggests.

H

2 : B is a sYmmetric matrix.

There is a third hypothesis that would be a considerable convenience if it

were true

3

-1, ~j=l _{b ij = 0}

for i

=

1, 2, 3 .

The theory supporting this model specification follows; the

reader who has

no interest in the theory can skip over the rest of the example.

(9)

6-1-4

the assumption of an ability to rank bundles is equivalent to the assumption

that there is a (utility) function u(q) such that u(qO)

>

u(q*)

~eans

bundle

qO is preferred to bundle q*.

Since a bundle costs p'q with p'= (Pl'P2, ... ,PN)

the consumers problem is

maximize u(q)

subject to p'q =

Y •

This is the same problem as

maximize u(q)

subject to

(p/Y)'q

=

1

which means that the solution must be of the form

q =

q(v)

with v

=

ply.

The function q(v) mapping the positive orthant of

RN

into the

positive orthant of R

N

is called the consumer's demand system.

It is usually

assumed in applied work that all prices are positive and that a bundle with

some q.

=

0 is never chosen.

1

If one substitutes the demand system q(v) back into the utility function

one obtains the function

g(v)

=

u[q(v)]

which gives the maximum utility that a consumer can achieve at the pricel

income point v.

The function g( v) is called the indirect utility function.

(10)

q(v)

=

(%v)g(v)/v'(%v)g(v).

This relationship is called Roy's identity.

Thus, to implement the theory

of consumer demand one need only specifY a parametric form g(vle) and then

fi t the system

q

=

(%v)g(vle)/v'(%v)g(vle)

to observed values of (q,v) in order to estimate e.

The theory asserts that

g(vle) should be decreasing in each argument,

and should be quasi-convex, v'(o2/ovov ')g(vle)v

>

0 the fitted function

(%v. )g(vle)

<

0 ,

~

every v with v'(%v)g(vle)

=

0 (Deaton and Muellbauer, 1980) .

for

If g(vle) has

this property then there exists a corresponding u(q).

Thus, in applied

work, there is no need to bother with u(q);

g(vle) is enough.

It is easier to arrive at a stochastic model if we reexpress the demand

system in terms of expenditure shares.

Accordingly let diag(v) denote a

diagonal matrix with the components of the vector v along the diagonal and set

s

=

diag( v)q

s(vle)

=

diag(v) (%v)g(vle)/v'(%v)g(vle) .

Observe that

s.

=

V.q.

=

p.q./y

~ ~ ~ ~ ~

so that s. denotes that proportion of total expenditure Y spent on the ith

~

good.

As such 1's

=

if

IS.

=

1

and 1 's(vle)

=

1 .

~= ~

(11)

location parameter

IJo =

tn s ( v

Ie)

where tn s(vle) denotes the N-vector with components

tn

s.(vle) for

l

i

= 1, 2, ... ,

N. The logistic-normal distribution is characterized as

follows.

Let w be normally distributed with mean vector

IJo

and a

variance-covariance matrix C(w,w') that satisfies 1 'C(w,w')1

=

O. Then s has the

logistic-normal distribution if

w.

where ew denotes the ve ctor with components e

l

for i

= 1, 2, ... ,

N.

A

log transform yields

whence

i = 1, 2, ... ,

N-l .

Writing wi - w

N

=

lJo

i

-

~ +

e

i

for i

=

1, 2, ... ,

N-l we have equations that

can be fi t to data

i=1,2, ...

,N-l.

The last step in implementing this model is the specification of a

(12)

or

g(v\e)

=

z!!

la.1.n(v.)

+

-21r.~ l~

lb .. 1.n(v.) 1.n(v.)

~= ~ ~ ~= J= ~J ~ J

g(vle)

=

a'x

+

(~)x'Bx

with x

=

1.n v and

Differentiation yields

(%v)g(vle)

=

[diag(v)r1:a

+

~

(B

+

B')x] .

One can see from this expression that B can be taken to be symmetric without

loss of generality.

With this assumption we have

(%v)g(v\e)

=

[diag(v)rl(a

+

Bx) .

Recall that

in general shares are computed as

s(vle)

=

diag(v) (%v)g(vle)/v'(%v)g(vla)

which reduces to

s(v\e)

=

(a

+

Bx)/1'(a

+

Bx)

in this instance.

Differenced log shares are

The model set forth in the beginning paragraphs of this discussion follows

fro~

the above equation.

The origins of hypotheses H

(13)

6-1-8

One notes, however, that we applied this model not to all goods

ql' q2' ... , qN and income Y but rather to three categories of electricity

expenditure - peak

=

ql' intermediate

=

~,

base

=

q3 - and to total

electricity expenditure E.

A (necessary and sUfficient) condition that

per-mits one to apply the theory of demand essentially intact to the electricity

subsystem, as we have done, is that the utility function is of the form

(Blacorby, Primont, and Russell,

1978,

Ch.

5)

u[

u (1) (ql' q2' q3)' q4' ... , qN

J .

If the utility function is of this form and E is known it is fairly easy to

see that optimal allocation of E to ql' q2' and q3 can be computed by solving

maximize u(l)(ql' q2' q3)

subject to

i3

p.q.

=

E .

• 1 1 . 1 . 1.=

Since this problem has exactly the same structure as the original problem,

one just applies the previous theory with N

=

3 and Y

=

E

There is a problem in passing from the deterministic version of the

subsystem to the stochastic specification.

One usually prefers to regard

prices and income as independent variables and condition the analysis on p

and Y.

Expenditure in the subsystem, from this point of view, is to be

regarded as stochastic with a location parameter depending on p, Y and

possibly on demographic characteristics, viz

E

=

f(p, Y, etc.)

+

error.

(14)

that an analysis

conditi~ned

on E will be adequate.

In Chapter 8 we shall

present methods that take formal account of this problem.

In this connection, hypothesis H

3 implies that g(vle) is

ho~ogeneous

of degree one in v which in turn implies that the first-stage allocation

function has the form

f(p, Y, etc.)

=

ft.rr(pl' P2' P3)' P4' ... ,

~,Y,

etc.]

where rr(Pl' P2' P3) is a price index for electricity which must, itself, be

homogeneous of degree one in PI' P2' P3 (Blackorby, Primont, and Russell,

1978, Ch. 5).

This leads to major simplifications in the interpretation of

results which see Caves and Christensen (1980).

One word of warning regarding Table lc, all data is constructed following

the protocol described in Gallant and Koenker (1984) save income.

Some

income values have been imputed by prediction

fro~

a regression equation.

These values can be identified as those not equal to one of the values 500,

1500, 2500, 3500, 4500, 5500, 7000, 9000, 11000, 13500, 17000, 22500, 27500,

40000, 70711.

The listed values are the means of the questionnaire's class

boundaries save the last which is the mean of an open ended interval assuming

that income follows the Pareto distribution.

The prediction equation includes

variables not shown in Table lc, namely age and years of education of a

(15)

Table la. Household Electricity Expenditures by Time-of-Use, North Carolina, Average over Weekdays in July 1978.

Expenditure Share

t Treatment Base Intermediate Peak

Expenditure

(S per day)

1 1 0.056731 0.280382 0.662888 0.46931

2 1 0.103444 0.252128 0.644427 0.79539

3 1 0.158353 0.270089 0.571558 0.45756

4 1 0.108075 0.305072 0.586853 0.94713

5 1 0.083921 0.211656 0.704423 1.22054

6 1 0.112165 0.290532 0.597302 0.93181

7 1 0.071274 0.240518 0.688208 1.79152

8 1 0.076510 0.210503 0.712987 0.51442

9 1 0.066173 0.202999 0.730828 0.78407

10 1 0.094836 0.270281 0.634883 1.01354

1 1 1 0.078501 0.293953 0.627546 0.83854

12 1 0.059530 0.228752 0.711718 1.53957

13 1 0.208982 0.328053 0.462965 1.06694

14 1 0.083702 0.297272 0.619027 0.82437

15 1 0.138705 0.358329 0.502966 0.80712

16 1 0.111378 0.322564 0.566058 0.53169

17 1 0.092919 0.259633 0.647448 0.85439

18 1 0.039353 0.158205 0.802442 1.93326

19 1 0.066577 0.247454 0.685970 1.37160

20 2 0.102844 0.244335 0.652821 0.92766

21 2 0.125485 0.230305 0.644210 1.80934

22 2 0.154316 0.235135 0.610549 2.41501

23 2 0.165714 0.276980 0.557305 0.84658

24 2 0.145370 0.173112 0.681518 1.60788

25 2 0.184467 0.268865 0.546668 0.73838

26 2 0.162269 0.280939 0.556792 0.81116

27 2 0.112016 0.220850 0.667133 2.01503

28 2 0.226863 0.257833 0.515304 2.32035

29 2 0.118028 0.219830 0.662142 2.40172

30 2 0.137761 0.345117 0.517122 0.57141

3 1 2 0.079115 0.257319 0.663566 0.94474

32 2 0.185022 0.265051 0.549928 1.63778

33 2 0.144524 0.276133 0.579343 0.75816

34 2 0.201734 0.241966 0.556300 1.00136

35 2 0.094890 0.227651 0.677459 1.11384

36 2 0.102843 0.264515 0.632642 1.07185

37 2 0.107760 0.214232 0.678009 1.53659

38 2 0.156552 0.236422 0.607026 0.24099

39 2 0.088431 0.222746 0.688822 0.58066

40 2 0.146236 0.301884 0.551880 2.52983

41 3 0.080802 0.199005 0.720192 1.14741

42 3 0.100711 0.387758 0.511531 0.97934

43 3 0.073483 0.335280 0.591237 1.09361

44 3 0.059455 0.259823 0.680722 2.19468

45 3 0.076195 0.378371 0.545434 1.98221

(16)

Table la. (Continued) .

Expenditure Share

t Trea.tment Base Intermediate Peak

Expenditure

($ per day)

46 3 0.076926 0.325032 0.598042 1.78194

47 3 0.086052 0.339653 0.574295 3.24274

48 3 0.069359 0.278369 0.652272 0.47593

49 3 0.071265 0.273866 0.654869 1.38369

50 3 0.100562 0.306247 0.593191 1.57831

51 3 0.050203 0.294285 0.655513 2.16900

S2 3 0.059627 0.311932 0.628442 2.11575

53 3 0.081433 0.328604 0.589962 0.35681

54 3 0.075762 0.285972 0.638265 1.55275

55 3 0.042910 0.372337 0.584754 1.06305

56 3 0.086846 0.340184 0.572970 4.02013

57 3 0.102537 0.335535 0.561928 0.60712

58 3 0.068766 0.310182 0.620452 1.15334

59 3 0.058405 0.307111 0.634485 2.43191

60 4 0.055227 0.300839 0.643934 0.10082

61 4 0.107435 0.273937 0.618628 0.69302

62 4 0.105958 0.291205 0.602837 1.12592

63 4 0.132278 0.219429 0.588293 1.84425

64 4 0.094195 0.328866 0.576940 1 .57972

65 4 0.115259 0.401079 0.483663 1.27034

66 4 0.150229 0.317866 0.531905 0.56330

67 4 0.168780 0.307669 0.523551 3.43139

68 4 0.118222 0.318080 0.563698 1.00979

69 4 0.103394 0.301611 0.588936 2.08458

10 4 0.124001 0.362115 0.513819 1.30410

11 4 0.197987 0.280130 0.521884 3.48146

72 4 0.108083 0.337004 0.554913 0.53206

13 5 0.088798 0.232568 0.618634 3.28981

74 5 0.100508 0.272139 0.627353 0.32678

75 5 0.127303 0.298519 0.574178 0.52452

76 5 0.109718 0.228172 0.662109 0.36622

77 5 0.130080 0.231037 0.638883 0.63788

78 5 0.148562 0.323579 0.521859 1.42239

79 :5 0.106306 0.252137 0.641556 0.93535

80 5 0.080877 0.214172 0.704951 1.26243

81 5 0.081810 0.135665 0.782525 1.51472

82 5 0.131749 0.278338 0.589913 2.07858

83 5 0.059180 0.254533 0.686287 1.60681

84 5 0.078620 0.267252 0.654128 t.54706

85 5 0.090220 0.293831 0.615949 2.61162

86 5 0.086916 0.193967 0.719117 2.96418

81 5 0.132383 0.230489 0.637127 0.26912

88 5 0.085560 0.252321 0.662120 0.42554

89 5 0.071368 0.276238 0.652393 1.01926

90 S 0.061196 0.245025 0.693780 1.53801

(17)

Expenditure Share

Expenditure

($ per day)

91 5 0.086608 0.233981 0.679411 0.75711

92 5 0.105628 0.,305471 0.588901 0.83647

93 5 0.078158 0.202536 0.719307 1.92096

94 5 0.048632 0.216807 0.734560 1.57795

95 5 0.094527 0.224344 0.681128 0.83216

96 5 0.092809 0.209154 0.698037 1.39364

97 5 0.035751 0.166231 0.798018 1.72697

98 5 0.065205 0.205058 0.729736 2.04120

99 5 0.092561 0.193848 0.713591 2.04708

100 5 0.063119 0.234114 0.702767 3.43969

101 5 0.091186 0.224488 0.684326 2.66918

102 5 0.047291 0.262623 0.690086 2.71072

103 5 0.081575 0.206400 0.712025 3.36803

104 5 0.108165 0.243650 0.648185 0.65682

lOS 5 0.079534 0.320450 0.600017 0.95523

106 5 0.084828 0.247189 0.667984 0.61441

107 5 0.063747 0.210343 0.725910 1.85034

108 5 0.081108 0.249960 0.668932 2.11274

109 5 0.089942 0.206601 0.703457 1.54120

110 5 0.046717 0.224784 0.728499 3.54351

111 5 0.114925 0.272279 0.612796 2.61769

112 5 0.115055 0.264415 0.620530 3.00236

113 S 0.081511 0.223870 0.694618 1.74166

114 5 0.109658 0.343593 0.546750 1.17640

115 5 0.114263 0.304761 0.580976 0.74566

116 5 0.115089 0.226412 0.658499 1.30392

117 5 0.040622 0.198986 0.760392 2.13339

118 5 0.073245 0.238522 0.688234 2.83039

119 5 0.087954 0.287450 0.624596 1.62179

120 5 0.091967 0.206131 0.701902 2.18534

121 5 0.142746 0.302939 0.554315 0.26503

122 5 0.117972 0.253811 0.628217 0.05082

123 5 0.071573 0.248324 0.680103 0.42740

124 5 0.073628 0.290586 0.635786 0.47979

125 5 0.121075 0.350781 0.528145 0.59551

126 5 0.077335 0.339358 0.583307 0.47506

127 5 0.074766 0.167202 0.758032 2.11867

128 5 0.208580 0.331363 0.460058 1.13621

129 5 0.080195 0.210619 0.709185 2.61204

130 5 0.066156 0.204118 0.729726 1.45227

131 5 0.112282 0.252638 0.635080 0.79071

132 5 0.041310 0.093106 0.865584 1.30697

133 5 0.102675 0.297009 0.600316 0.93691

134 5 0.102902 0.270832 0.626266 0.98718

135 5 0.118932 0.250104 0.630964 1.40085

(18)

Expenditure Share

t Treatment Base Intermediate Peale

Expenditure

($ per dal{)

136 5 0.139760 0.322394 Q.537846 1.78710

137 5 0.121616 0.214626 0.663758 8.46237

138 5 0.065701 0.263818 0.670481 1.58663

139 5 0.034029 0.175181 0.790790 2.62535

140 5 0.074476 0.194744 0.730780 4.29430

141 5 0.059568 0.229705 0.710727 0.65404

142 5 0.088128 0.295546 0.616326 0.41292

143 5 0.075522 0.213622 0.710856 2.02370

144 5 0.057089 0.195720 0.747190 1. 76998

145 5 0.096331 0.301692 0.601977 0.99891

146 5 0.120824 0.250280 0.628896 0.27942

147 6 0.034529 0.193456 0.772015 0.91673

148 6 0.026971 0.180848 0.792181 1.15617

149 6 0.045271 0.141894 0.812835 1.57107

150 6 0.067708 0.219302 0.712990 1.24515

151 6 0.079335 0.230693 0.689972 1.70748

152 6 0.022703 0.178896 0.798401 1.79959

153 6 0.043053 0.157142 0.799805 4.61665

154 6 0.057157 0.245931 0.696912 0.59504

155 6 0.063229 0.136192 0.800579 1.42499

156 6 0.076873 0.214209 0.708918 1.34371

157 6 0.027353 0.124894 0.847753 2.74908

158 6 0.067823 0.146994 0.785183 1.84628

159 6 0.056388 0.189185 0.754428 3.82472

160 6 0.036841 0.194994 0.768165 1.18199

161 6 0.059160 0.138681 0.802158 2.07338

162 6 0.051980 0.215700 0.732320 0.80376

163 6 0.027300 0.145072 0.827628 1.52316

164 6 0.014790 0.179619 0.805591 3.17526

165 6 0.047865 0.167561 0.784574 3.30794

166 6 0.115629 0.231381 0.652990 0.72456

167 7 0.104970 0.147525 0.747505 0.50274

168 7 0.119254 0.187409 0.693337 1.22571

169 7 0.042564 0.112839 0.844596 2.13534

170 7 0.096756 0.150178 0.753066 5.56011

171 7 0.063013 0.168422 0.768565 3.11725

172 7 0.080060 0.143934 0.776006 0.99796

173 7 0.097493 0.173391 0.729116 0.67859

174 7 0.102526 0.220954 0.676520 0.79027

175 7 0.085538 0.195686 0.718776 2.24498

176 7 0.068733 0.166248 0.765019 2.01993

177 7 0.094915 0.140119 0.764966 4.07330

178 7 0.076163 0.132046 0.791792 3.66432

179 7 0.099943 0.176885 0.723172 0.40768

180 7 0.081494 0.175082 0.743425 1.09065

(19)

Ta.ble la.. (Continued) .

Expenditure Share

6-1-14

Expenditure

($ per day)

181 7 0.196026 0.299348 0.504626 1.35008

182 7 0.093173 0.235816 0.671011 1.06138

183 7 0.172293 0.173032 0.654675 0.99219

184 7 0.067736 0.159600 0.772663 3.69199

185 7 0.102033 0.171697 0.726271 2.36676

186 7 0.067977 0.151109 0.780914 1.84563

187 8 0.071073 0.238985 0.689942 0.18316

188 8 0.049453 0.286788 0.663759 2.23986

189 8 0.062748 0.255129 0.682123 3.48084

190 8 0.032376 0.154905 0.812719 7.26135

191 8 0.055055 0.225296 0.719648 1.68814

192 8 0.037829 0.179051 0.783120 1.13804

193 8 0.020102 0.172396 0.807502 1.40894

194 8 0.021917 0.149092 0.828992 3.47472

195 8 0.047590 0.174735 0.777675 3.37689

196 8 0.063446 0.235823 0.700731 3.14810

197 8 0.034719 0.159398 0.805883 3.21710

198 8 0.055428 0.200488 0.744084 1.13941

199 8 0.058074 0.254823 0.687103 2.55414

200 8 0.060719 0.209763 0.729518 0.29071

201 8 0.045681 0.206177 0.748142 1.21336

202 8 0.040151 0.263161 0.696688 1.02370

203 8 0.072230 0.281460 0.646310 1.40580

204 8 0.064366 0.269816 0.665819 0.97704

205 8 0.035993 0.191422 0.772585 2.09909

206 9 0.091638 0.215290 0.693073 1.03679

207 9 0.072171 0.236658 0.691171 2.36788

208 9 0.056187 0.195345 0.748468 3.45908

209 9 0.095888 0.229586 0.674526 3.63796

210 9 0.069809 0.219558 0.710633 2.56887

211 9 0.142920 0.223801 0.633279 2.00319

212 9 0.087323 0.196401 0.716276 2.40644

213 9 0.064517 0.218711 0.716772 2.58552

214 9 0.086882 0.194778 0.718341 8.94023

215 9 0.067463 0.219228 0.713309 3.75275

216 9 0.105610 0.230661 0.663730 0.34082

217 9 0.138992 0.283123 0.577885 1.62649

218 9 0.081364 0.186967 0.731670 2.31678

219 9 0.114535 0.221751 0.663714 1.77709

220 9 0.069940 0.280622 0.649438 1.38765

221 9 0.073137 0.143219 0.783643 3.46442

222 9 0.096326 0.243241 0.660434 1.74696

223 9 0.083284 0.202951 0.713765 1.28613

224 9 0.179133 0.299403 0.521465 1.15897

(20)

Taule lb. Experimental Rates in Effect on a Veekday in July 1978.

Price (cents per kwh)

Treatment Base Intermedite Peak

1 1 .06 2.86 3.90

2 1.78 2 .86 3.90

3 1 .06 3.90 3.90

4 1 .78 3.90 3.90

5 1 .37 3.34 5.06

6 1 .06 2 .86 6 .56

7 1 .78 2.86 6.56

8 1 .06 3.90 6.56

9 1 .78 3.90 6.56

Base period hours are l1pm to 7am. Intermediate period hours are 7am to lOam and 8pm to llpm. Peak period hours

(21)

Table lc. Consumer Demographic Characteristics.

6-1-16

Residence

---

Air Condition.

Heat Elec.

---F am i 1y Income Size Loss Range \.lasher Dryer Central \.lindow t Size ($ per yr) (SqFt) (Btuh) (l=yes) (l=yes) (l=yes) (l=yes) (Btuh)

1 2 17000 600 4305 0 1 0 0 13000

2 6 13500 700 7731 1 1 0 0 0

3 2 7000 1248 18878 1 1 0 0 0

4 3 11000 1787 17377 1 1 0 0 0

5 4 27500 2700 24874 1 0 0 1 5000

6 3 13500 2000 ZZ5Z6 1 1 1 0 24000

7 4 22500 3800 17335 1 1 1 1 0

8 7 3060 216 4476 1 0 0 0 0

9 3 7000 1000 8772 0 1 1 0 18000

10 1 6773 1200 14663 0 0 0 0

11 5 11000 1000 14480 1 1 0 0 0

12 5 17000 704 3172 1 1 1 1 24000

13 3 5500 2100 8631 1 1 0 1 0

14 2 13500 1400 17720 1 1 1 0 17000

15 4 22500 1252 7386 1 1 1 0 24000

16 7 17000 716 7174 0 1 0 0 0

17 2 11000 1800 17757 1 1 1 1 0

18 2 13500 780 4641 1 1 0 1 0

19 3 6570 960 11396 1 1 0 0 24000

20 4 9000 768 8195 1 1 1 0 0

21 2 11000 1200 7812 1 1 1 1 10000

2Z 4 13500 900 8878 1 1 1 1 0

23 3 40000 2200 15078 1 1 1 0 0

24 5 7000 1000 7041 1 1 0 0 10000

25 3 13500 720 5130 0 1 1 0 0

Z6 2 13500 550 7532 1 1 0 0 12000

27 4 17000 1600 9674 1 1 1 1 0

28 4 27500 2300 13706 1 1 0 1 0

29 6 15777 1000 10372 1 1 1 0 10000

30 2 11000 880 7477 0 1 1 0 17000

31 4 9000 1200 14013 1 1 1 0 0

32 4 17052 2200 15230 1 1 0 0 0

33 2 14812 1080 13170 1 0 0 0 0

34 3 27500 870 10843 1 1 1 0 18500

35 2 4562 800 9373 1 1 1 0 6000

36 2 7000 1200 11395 1 1 0 0 0

37 3 9000 700 6175 1 1 0 0 23000

38 2 4711 1500 17655 1 0 0 0 0

37 5 146 52 1500 11916 1 1 1 0 0

40 4 70711 2152 16552 1 1 1 1 0

41 2 7000 832 4316 1 1 1 1 0

42 3 22500 1700 7209 1 1 1 1 0

43 11 4500 1248 7607 1 1 0 0 0

44 5 11000 1808 19400 1 1 1 0 28000

45 6 22500 1800 177 81 1 1 1 1 0

(22)

Residence

---

Air Condition.

Heat Elec.

---F am i I Y Income Size Loss Range lJasher Dryer Central lJindow Si ze ( $ per yr) (SqFt) (Stuh) (l=yes) (l=yes) (l=yes) (l=yes) (Stuh)

46 4 22500 1800 18573 0 0 0 1 0

47 3 40000 4200 16264 1 1 1 1 0

48 2 9000 1400 10541 1 1 1 0 24000

49 2 13500 2500 29231 1 1 0 0 16000

SO 6 17000 1300 5805 1 1 1 0 21000

51 3 11000 780 5894 1 1 1 1 0

52 1 4500 1000 13714 0 0 0 0 6000

53 2 11267 960 7863 1 1 0 0 0

54 3 2500 1000 12973 1 1 0 0 0

55 1 7430 1170 9361 1 1 1 0 0

56 4 17000 2900 12203 1 1 1 1 0

57 1 22500 1000 10131 0 1 0 0 0

58 3 22500 1250 12773 1 1 1 0 12000

59 3 7000 1400 11011 1 1 1 0 29000

60 1 2500 835 12730 1 0 0 0 0

61 1 13500 1300 1196 1 1 0 0 32000

62 1 11 000 540 1198 1 1 0 0 0

63 4 14381 1100 8100 1 1 1 0 30000

64 2 9000 900 5126 1 0 0 0 12000

65 3 11000 120 3854 1 1 1 1 0

66 5 5500 180 6236 1 1 0 1 0

67 4 40000 1450 8160 1 1 1 0 28000

68 2 3500 1100 10102 1 1 0 0 12000

69 2 11000 3000 36124 1 1 0 1 0

70 4 11000 1534 15711 1 0 0 0 0

11 2 40000 2000 11250 1 1 1 1 0

72 2 2500 1400 15040 0 0 0 0 6000

73 4 11000 1400 13544 1 0 1 1 0

14 2 1500 656 1383 1 0 0 0 0

15 3 9000 712 13229 1 0 0 0 1800

76 1 9000 600 4035 1 1 0 0 0

77 5 5500 500 6110 1 0 0 0 0

78 3 13500 1200 11097 1 1 1 0 10000

19 2 13590 1300 12869 1 0 0 0 24000

80 4 11000 1045 11224 1 1 0 0 0

81 2 9681 768 1565 1 1 1 0 10000

82 2 17000 1100 9159 0 1 1 0 10000

83 11 4500 480 6099 1 1 0 0 0

84 5 13500 1916 12478 1 1 1 0 0

85 4 40000 2500 23213 1 1 1 0

86 5 22500 2100 12314 1 1 1 1 0

81 3 3500 1196 14125 0 0 0 0 0

88 3 12100 950 11114 0 0 0 0 0

89 3 3500 1080 12186 1 0 0 0 0

90 2 1000 1400 10050 1 1 0 0 28000

(23)

Table lc. (Continued).

Residence

---

Air Condition.

Heat Elec.

---Fam i 1Y Income Size Loss Range Washer Dryer Central Window t Size ($ per yr) (SqFt) (Stuh) (l=yes) (l=yes) (l=yes) (l=yes) (Stuh)

91 2 3500 1800 16493 1 1 1 0 2000

92 2 7000 1456 17469 0 1 0 0 18000

93 4 9000 1100 6177 1 1 1 0 23000

94 2 3500 1500 21659 1 1 1 0 18000

95 4 9894 720 6133 1 1 1 0 6000

96 1 22500 1500 7952 1 0 0 1 0

97 4 13500 1500 10759 1 0 1 1 0

98 4 17000 1900 10176 1 1 1 1 0

99 2 17000 1100 10869 1 1 1 0 23000

100 5 27500 2300 16610 1 1 1 1 0

101 3 13500 1500 11304 1 1 1 1 0

102 2 27500 3000 23727 1 1 1 1 0

103 4 24970 2280 18602 1 1 1 1 0

104 2 3500 970 10065 1 1 0 0 0

105 2 17000 1169 1081 0 1 1 0 0 30000

106 2 13500 1800 20614 1 1 1 0 0

107 2 13500 728 4841 1 1 1 1 0

108 2 11000 1500 11235 1 1 1 1 0

109 3 17000 1500 9774 1 1 0 1 0

110 5 5500 900 12085 1 1 0 0 23000

111 3 17000 1500 17859 1 1 1 1 0

112 1 70711 2600 16 661 1 1 1 1 0

113 3 7000 780 5692 1 1 1 0 20000

114 4 22500 1600 8191 1 1 1 1 0

115 2 13500 600 5086 0 1 1 0 2000

116 3 4500 1200 14178 1 1 1 0 1000

117 5 17000 900 8966 1 1 1 0 18000

118 4 13500 1500 11142 1 1 1 1 0

119 5 17000 2000 19555 1 1 1 1 0

120 3 23067 1740 10183 1 1 1 0 42000

121 1 17000 696 5974 1 0 0 0 0

122 1 2500 900 10111 1 1 0 0 0

123 2 7265 970 20437 1 1 0 0 0

124 2 10415 1500 9619 1 0 0 0 0

125 3 5500 750 169 55 0 0 1 0 18000

126 2 4500 824 11647 1 1 0 0 0

127 1 22500 1900 11401 1 0 1 1 0

128 4 40000 2500 15205 1 1 1 1 0

129 2 4500 840 5984 1 1 1 1 0

130 1 22500 1800 18012 1 1 1 1 0

131 2 5500 1200 8447 1 1 0 0 1000

132 1 3689 576 12207 0 0 0 0 0

133 3 16356 1600 16227 0 1 1 0 28500

134 4 11000 1360 17045 1 1 0 0 0

135 3 5500 600 4644 0 1 0 0 9000

(24)

Residence

---

Air Condition.

Heat Elec.

---F am i I Y Income Sise Loss Range Washer Dryer Central Window t Sise ($ per yr) (SqFt) (Btuh) _(l=yes) _(l=yes) _{(1=1 es ) (1=1es)} _{(B tuh)}

136 3 17000 2000 16731 1 1 1 1 2300

137 2 32070 6000 61737 1 1 1 1 0

138 2 27500 1250 7397 1 1 1 1 0

139 4 17000 840 5426 1 1 1 1 0

140 4 27500 3300 11023 1 1 1 1 0

141 2 11000 1200 10888 1 0 0 0 18000

142 1 1000 5446 1 0 0 0 0

143 3 36919 1200 8860 1 1 1 1 0

144 5 9000 720 5882 1 1 1 0 10000

145 5 21400 1300 6273 1 1 1 0 0

146 1 1500 375 6727 0 0 0 0 0

147 2 5063 1008 7195 1 0 0 0 0

148 1 3500 1650 13164 1 0 0 1 0

149 1 9488 850 9830 0 0 1 0 10000

150 1 27500 1200 8469 1 1 1 1 0

151 5 17000 1000 8006 0 1 1 0 16 0 00

152 3 11000 2000 12608 1 1 1 1 0

153 7 22500 1225 11505 1 0 0 1 0

154 6 3500 1200 16682 1 1 0 0 0

155 3 9273 600 5078 1 1 0 0 15000

156 8 17000 1100 17912 1 0 0 0 0

157 3 17459 980 7984 0 1 1 1 0

158 5 11000 1200 14113 1 1 1 0 18000

159 3 9000 1600 21519 1 1 1 0 6000

160 2 11000 899 5731 0 1 1 0 28000

161 3 12068 1350 16331 1 1 1 0 6000

162 2 7000 672 8875 1 1 0 0 0

163 3 22500 1200 10424 1 1 0 0 23000

164 2 5500 1300 8636 1 1 1 1 0

165 2 12519 1000 24210 1 1 1 0 37000

166 2 29391 1400 12837 1 1 1 1 0

167 2 9000 400 4519 1 0 0 0 0

168 3 4664 1235 14274 1 1 0 0 6000

169 4 11000 720 6393 0 1 1 0 23000

170

171 3 18125 2300 16926 1 0 1 0

172

173 5 9000 720 6439 1 1 1 0 0

174 6 5500 1000 13651 1 1 0 0 0

175 5 14085 1400 14563 1 1 0 0 15000

176 2 9000 720 6540 0 1 1 1 0

177 6 17000 1470 8439 1 1 1 1 0

178 4 27500 1900 12345 1 1 1 1 18500

179 3 7000 480 3796 0 0 0 0 10000

180 3 13500 1300 7352 1 1 0 0 23000

(25)

T~ble lc. (Continued).

Residence

---

Air Condition.

Heat Elec.

---F~mi I Y Income Size Loss R~nge "'asher Dryer Central "'indow t Sin ($ per yr) (SqFt) (Stuh) (l=yes) ( l=yes) (l=yes) (1=ye5) (Stuh)

181 3 13437 1200 9502 1 1 1 1 0

182 3 14150 1300 8334 1 1 0 0 0

183 1 7000 1200 119 41 1 1 0 0 21 00 0

184 4 27500 1350 7585 1 1 1 1 0

185 2 32444 2900 15158 1 1 0 1 0

186 1 4274 400 7859 1 0 0 0 0

187 1 3500 600 144 41 0 0 0 0 0

188 4 27500 2000 15462 1 1 1 1 0

189 4 40000 2900 13478 1 1 0 1 0

190 6 17000 5000 24132 1 0 1 1 0

191 1 2500 1400 17016 1 1 0 0 2000

192 7 9000 1400 13293 1 1 0 0 0

193 0 0 0 0

194 4 13500 780 5629 1 1 0 1 0

195 5 13500 1000 7281 1 1 1 1 0

196 2 13500 1169 11273 1 1 0 0 12000

197 2 40000 2400 13515 1 1 0 1 0

198 4 27500 1320 9865 1 1 1 0 29000

199 4 27500 1250 5759 1 1 1 1 0

200 1 3449 1200 18358 0 0 0 0 0

201 2 3500 425 4554 1 0 0 0

202 2 27500 1400 13496 1 0 0 1 0

203 4 7000 1300 11555 1 1 1 0 14000

204 2 3500 1800 23271 1 1 0 0 0

205 4 11000 720 5879 1 1 1 0 16000

206 7 9000 680 11528 1 0 0 0 0

207 4 14077 780 4829 1 1 1 0 10000

208 3 13500 2200 22223 1 1 1 0 24000

209 4 17000 1342 12050 1 1 1 1 0

210 4 3500 628 5369 1 1 1 0 24000

211 2 11000 920 5590 1 1 1 1 0

212 :5 9000 1300 11510 1 1 1 0 19000

213 3 5500 1400 18584 1 1 1 0 23000

214 5 27500 2300 15480 1 1 1 1 0

215 3 20144 1700 11212 1 1 1 1 0

216 5 3500 1080 13857 0 0 0 0 0

217 2 22500 1800 17588 1 1 0 0 23000

218 6 22500 1900 15115 1 1 1 0 22000

219 5 6758 1200 16868 1 0 0 0 0

220 6 11000 2200 21884 1 1 1 1 0

221 3 17000 1500 11504 1 1 1 1 0

222 2 9000 600 5825 1 0 1 1 0

223 2 15100 1932 15760 1 1 1 0 0

224 1 7000 979 11700 1 1 1 0 1000

(26)

Type of Residence

---

Elec.

DupIn or Mobile Water

Detached Apartment 'Home Heater Freezer Refrigerator t (1=yes) (1=yes) (1=yes) (1=yes) ( kw) ( kw)

1 0 0 1 1 0 0.700

2 1 0 0 1 1.320 0.700

3 1 0 0 1 1 .320 0.700

4 1 0 0 1 1.320 2.495

5 1 0 0 0 1 .320 3.590

6 1 0 0 1 0 1.795

7 1 0 0 1 0 1 .795

8 1 0 0 0 1.320 0.700

9 1 0 0 1 1 .320 0.700

10 1 0 0 0 1 .985 1.795

11 1 0 0 1 2.640 0.700

12 0 0 1 1 1.985 1.795

13 1 0 0 1 1.320 1 .795

14 1 0 0 1 1.320 1.795

15 1 0 0 1 1.320 1 .795

16 0 0 1 1 0 1.795

17 1 0 0 1 1 .985 1 .795

18 0 0 1 1 0 0.700

19 1 0 0 1 1.320 1 .795

20 1 0 0 1 0 1.795

21 1 0 0 1 1.320 0.700

22 1 0 0 1 1.320 0.700

23 1 0 0 1 3.305 1 .795

24 0 1 0 1 0 0.700

25 0 0 1 1 1.320 0.700

26 1 0 0 1 1.985 0.700

27 1 0 0 1 1 .320 1 .795

28 1 0 0 1 1 .320 1.795

29 1 0 0 1 0 1 .795

30 0 0 1 1 0 0.700

31 1 0 0 1 1 .320 0.700

32 1 0 0 1 0 1.795

33 1 0 0 1 1 .320 3.590

34 1 0 0 1 0 1.795

35 1 0 0 1 1 .320 1 .795

36 1 0 0 1 1 .320 0.700

37 1 0 0 1 1 .320 1 .795

38 1 0 0 1 0 0.700

39 1 0 0 1 0 1 .795

40 1 0 0 1 1 .320 1.400

41 1 0 0 1 1 .985 1 .795

41 1 0 0 1 2.640 0.700

43 1 0 0 1 1.320 1 .795

44 1 0 0 1 3.970 1.795

45 1 0 0 1 1 .9 B5 1 .795

(27)

Table Ie. (Continued).

Type of Residence

---

Elec.

DupIn or Mo biIe 'Jater

Detached Apartment Home Heater Freezer Refrigerator t (1=yes) (1=1 es ) (1=yes) (1=ye5) ( kw) (kw)

46 1 0 0 0 0 1 .795

47 1 0 0 1 0 1.795

48 1 0 0 1 0 0.700

49 1 0 0 1 0 2.495

50 1 0 0 1 1 .985 1 .795

51 0 0 1 1 1.985 0.700

52 1 0 0 0 1 .320 1 .795

53 0 0 1 1 0 0.700

54 1 0 0 1 1 .320 1 .795

5S 1 0 0 1 1.320 1.795

56 1 0 0 1 1 .320 1 .795

57 1 0 0 1 1.320 1.795

58 1 0 0 1 1 .320 1 .795

59 1 0 0 1 1.320 1.795

60 1 0 0 0 0 0.700

61 1 0 0 1 1 .320 0.700

6Z 1 0 0 1 1 .320 0.700

63 1 0 0 1 1.320 0.700

64 0 1 0 1 0 1 .795

6S 0 0 1 1 0 0.700

66 0 0 1 1 0 1 .795

67 1 0 0 1 1 .320 1.795

68 1 0 0 1 1.320 0.700

6? 1 0 0 1 0 1.400

70 1 0 0 1 1 .320 1 .795

71 1 0 0 1 1 .985 1.795

72 1 0 0 0 1.320 1 .795

73 1 0 0 0 0 1.795

74 1 0 0 0 0 0.700

7S 1 0 0 0 1.320 0.700

76 1 0 0 1 0 0.700

77 0 1 0 0 1 .320 0.700

78 1 0 0 1 0 1 .795

79 1 0 0 0 1.320 1.795

80 1 0 0 1 1.320 1 .795

81 0 0 1 1 1.320 2.495

82 1 0 0 1 1 .985 1 .795

83 1 0 0 1 1 .320 0.700

84 1 0 0 1 1 .985 1 .795

85 1 0 0 1 0 1.795

86 1 0 0 1 1 .320 2.495

87 1 0 0 0 0 1.795

88 1 0 0 0 3.305 0.700

89 1 0 0 1 1 .985 0.700

90 1 0 0 1 1 .985 1 .795

(28)

Type of Residence

---

Elec.

Duplex or Mobile Water

Detached Apartment Home Heater Freezer Refrigerator t (l=yes) (l=yes) (l=yes) (1=ye5) ( kw) <kw)

91 1 0 0 1 0 1.795

92 1 0 0 1 1.985 1.795

93 1 0 0 1 1 .320 1 .795

94 1 0 0 1 0 1.795

95 0 0 1 1 0 1 .795

96 1 0 0 1 1 .985 0.700

97 1 0 0 0 1.320 0.700

98 1 0 0 1 1.320 0.700

99 1 0 0 1 2.640 1 .795

100 1 0 0 1 1 .320 1.795

101 1 0 0 1 1 .320 1 .795

102 1 0 0 1 0 2.495

103

_I

0 0 1 1 .320 1.795

104 1 0 0 1 1 .320 0.700

105 1 0 0 1 1.320 0.700

106 1 0 0 1 0 1.795

107 1 0 0 1 1.320 0.700

108 1 0 0 1 1.320 0.700

109 1 0 0 1 0 1 .795

110 1 0 0 1 1 .320 0.700

111 1 0 0 1 1 .320 1 .795

112 1 0 0 1 3.970 1.795

113 0 0 1 1 0 1 .795

114 1 0 0 1 1.320 1.795

115 0 0 1 1 0 0.700

116 1 0 0 1 0 1.795

117 1 0 0 1 1.320 1 .795

118 1 0 0 1 1.985 0.700

119 1 0 0 1 0 0.700

120 0 0 1 1 1.320 1.795

121 0 0 1 1 0 1 .795

122 1 0 0 1 0 0.700

123 1 0 0 1 0 0.700

124 1 0 0 1 0 1.795

125 1 0 0 0 1 .320 1 .795

126 1 0 0 1 0 0.700

127 1 0 0 0 1 .320 1.795

128 1 0 0 1 0 2.495

129 0 0 1 1 1 .320 1 .795

130 1 0 0 1 0 1.795

131 1 0 0 1 0

o

.700

132 1 0 0 0 1.320 1.795

133 1 0 0 1 1.320 0.700

134 1 0 0 1 1.320 0.700

135 0 0 1 1 0 0.700

(29)

Table Ie. (Continued).

6-1-24

Type of Residence

---

Elec.

Duplex or Mobile 'Water

Detached Apartment Home Heater Freezer Refrigerator t (l=yes) (l=yes) (l=yes) (1=ye5) ( kw) (kw)

136 1 0 0 1 1.985 1.795

137 1 0 0 1 1.985 1.400

138 0 1 0 1 0 1 .795

139 0 0 1 1 0 0.700

140 0 1 0 1 7.265 1 .195

141 1 0 0 1 1.320 0.700

142 0 1 0 1 0 1 .195

143 0 1 0 1 0 0.700

144 0 0 1 1 0 0.700

145 0 0 1 1 0 1.795

146 1 0 0 0 0 1 .795

147 1 0 0 0 0 0.700

148 1 0 0 0 0 1 .795

149 1 0 0 0 1 .320 1.795

150 0 1 0 1 0 1 .795

151 1 0 0 1 2.640 1.795

152 1 0 0 1 0 1.195

153 1 0 0 0 1 .320 1.195

154 1 0 0 1 1.320 0.700

155 0 0 1 1 0 0.700

156 1 0 0 1 1 .320 0.700

157 0 0 1 1 1.320 1.795

158 1 0 0 1 3.970 2.495

159 1 0 0 1 0 0.700

160 1 0 0 1 1 .320 1 .795

161 1 0 0 1 1.320 1.795

162 1 0 0 1 1 .320 0.700

163 1 0 0 1 0 1.795

164 1 0 0 1 0 1 .795

165 1 0 0 1 2.640 3.590

166 1 0 0 1 1 .320

o

.700

167 0 1 0 1 0 0.700

168 1 0 0 1 2.040

o

.700

169 0 0 1 1 0 1.795

170

171 1 0 0 1.320 1.795

172

173 0 0 1 1 1 .320 0.700

174 1 0 0 1 1.320 0.700

175 1 0 0 1 1.985 1.795

176 0 0 1 1 0 0.700

177 1 0 0 1 3.970 1.195

178 1 0 0 1 1 .985 1 .795

179 0 0 1 0 0 0.700

180 1 0 0 1 1.320 1 .195

(30)

Type of Residence

---

Elec.

Duplex or Mobile 'Ja.ter

Detached Aputmen t Home Heater Freezer Refrigerator

t (l=yes) (l=yes) (l=yes) (l=yes) (kw) (kw)

181 1 0 0 1 1.985 0.100

182 1 0 0 1 1.985 1.795

183 1 0 0 1 1.320 1 .795

184 1 0 0 1 1.320 1.795

185 1 0 0 1 0 2.495

186 1 0 0 1 0 1.795

187 1 0 0 0 1.320 0

188 1 0 0 1 1.320 1.795

189 1 0 0 1 0 1 .795

190 1 0 0 0 1 .985 2.495

191 1 0 0 1 0 1 .795

192 1 0 0 1 0 1.795

193 0 0 1 0 0 0

194 0 0 1 1 1.320 1.795

195 1 0 0 1 1 .320 0.700

196 1 0 0 1 1.985 1.795

197 1 0 0 1 1 .985 1 .795

198 0 0 1 1 0 1.795

199 1 0 0 1 1 .320 1 .795

200 1 0 0 0 0 0.700

201 0 1 0 1 1.985 0

202 0 1 0 1 0 1.795

203 1 0 0 1 1.320 0.700

204 1 0 0 1 0 0.700

205 0 0 1 1 1 .320

o

.700

206 1 0 0 0 1.320 0.700

207 0 0 1 1 0 0.700

208 1 0 0 1 1.320 2.495

209 1 0 0 1 1 .320 1 .195

210 1 0 0 1 0 0.700

211 0 1 0 1 0 0.700

212 1 0 0 1 0 1 .795

213 1 0 0 1 1.320 1 .795

214 1 0 0 1 1.985 1 .795

215 1 0 0 1 1.320 1 .795

216 1 0 0 0 0 0.700

217 1 0 0 1 1.320 1 .795

218 1 0 0 1 1.985 1.795

219 1 0 0 0 0

o

.700

220 1 0 0 1 1 .320 1.795

221 1 0 0 1 1 .320 1 .795

222 0 1 0 1 0 1.795

223 1 0 0 1 1.320 1 .795

224 1 0 0 1 0 1 .79S

(31)

2. LEAST SQUARES ESTIMATORS AND MATTERS OF NOTATION

Univariate responses y Ctt for t = 1, 2, ... , nand

Ct

= 1, 2, ... , M

are presumed to be related to k-dimensional input vectors x

t

as follows

Ct 1,2, ... ,M; t = 1 , 2 , ...

, n

where each f (x,S ) is a known function, each SO is a p -dimensional vector

Ct Ct Ct Ct

of unknown parameters, and the e

Ctt

represent unobservable observational or

experimental errors.

As previously, we write

~

to emphasize that it is the

Ct

true, but unknown, value of the parameter vector

e

Ct

that is meant;

80'

itself

is used to denote instances when the parameter vector is treated as a variable.

Writing

e

=

t

the error vectors e

t

are assumed to be independently and identically distributed

with mean zero and unknown variance-covariance matrix

2: ,

!:

= C(e

t

,

e~)

t = 1, 2,

...

,

n ,

whence

r:

a

t = s

C(e

_at

, eSt) =

t

=I

s

with

0'0'6

denoting the elements of

l: .

In the Iitera,ture one finds two conventions for writing this model in

a vector form.

One emphasizes the fact that the model consists of M separate

univariate nonlinear regressions

y

=f(SO)+e

a a Ct a

(32)

with y

being an n-vector as described below and the other emphasizes the

et

multivariate nature of the data

t = 1, 2, ... , n

with Y

t

being an M-vector;

Simply to have labels to distinguish the two,

we shall follow Zellner (1962) and refer to the first notational scheme as

the "seemingly unrelated" (nonlinear regressions) structure the second as

the multivariate (nonlinear regression) structure.

Let us take these up in

turn.

The "seemingly unrelated" notational scheme follows the same conventions

used in Chapter 1.

Write

Yet1

Yet2

Y

=

et

n

Yetn

1 f (x

l

,6

)

et et

f (x

2 ,6 )

et et

f

(6 )

=

et et

f (x

,6

)

n

et

n

et

1 eetl

eet2

e

₌

et

e

n

om

1 In this notation, each regression is written as

0

(33)

6-2-3

with (Problem 1)

C(e

,e~) = (J t:t

I

.

ex 10' exjii n n

Denote the Jacobian of f (8 ) by

ex ex

F (8 ) = (a/a8')

f

(8 )

ex ex ex ex ex

which is of order n by p

.

Illustrating with Example 1 we have:

ex

EXAMPLE 1 (continued).

The independent variables are the logarithms of

expenditure normalized prices.

From Tables la arid lb we obtain a few instances

xl = tn[(3.90, 2.86, 1.06)/(0.46931)J' = (2.11747, 1.80731, 0.81476)'

x2

=

tn[(3·90, 2.86, 1.06)/(0.79539)J' = (1.58990, 1.27974, 0.28719)'

x20=

~n[(3.90,

2.86, 1.06)/(1.37160)J' = (1.04500,0.73484, -0.25771)'

x21= tn((3.90, 2.86, 1.78)/(0.92766)J' = (1.43607,1.12591,0.65170)'

x40= tn[(3.90, 2.86, 1.78)/(2.52983)J' = (0.43282, 0.12267, -0.35154)'

x41= tn[(3.90, 3·90, 1.06)/(1.14741)J' = (1.22347, 1.22347, -0.079238)'

X

224= tn[(6.56, 3·90, 1.78)/(1.15897)J' = (1.73346,1.21344,0.42908)' .

The vectors of dependent variables are for

ex =

1

~n(0.662888/0.05673l)

tn(0.644427/0.103444)

y

=

ex •

tn(0.521465/0.l79l33)

224

1 =

2.45829

1.82933

1.06851

(34)

and for

et =

2 Ln(0.280382/0.056731)

Ln(0.252128/0.103444)

1.59783

0.89091

y

=

et

=

Ln(0.299403/0. 179133)

224

1 0.51366

224

1 Recall that

with beet) denoting the et-th row of

oa and

B

=

0

B we shall have

Note that if both a and B are multiplied by some

b

ll

b

12 b

13 B

=

b

21 b

22 b

23 b

31 b

32 b

33 and with a'

=

(a

l

, a

_{2 , a3}

)

.

co:nm.on factor

0

to obtain

a

=

Thus the parameters of the model can only be determined to within a scalar

multiple.

In order to estimate the model it is necessary to impose a

normal-ization rule.

Our choice is to set

8

3 =

-1.

With this choice we write the

model as

(35)

f (x,8 )

=

J.nl

(a

+

b' (a)x)/(-l

+

b(3)x)]

a

=

1, 2

a

8'

=

(a

l

, b

ll

, b

12,

b

13 , b

3l

, b

32 , b

33 )

1 e'

=

(a

2 , b

2l

, b

22 , b

23 , b

3l

, b

32 , b

33 )

0

2 Recognizing that what we have is

M

instances of the univariate nonlinear

regression model of Chapter 1, we can apply our previous results and estimate

the parameters

8°

of each model by computing

~#

to :ninimize

a

SSE

(8 )

=

[y - f

(8 )]

'L

y - f

(8 )]

a

for a

=

1, 2, ... ,

M.

This done, the elements

<J

a6

of

I:

can be estimated by

a,S

=

1, 2, ... , M .

Let

~

denote the M by M matrix with typical element

Cr

aS

.

Equi valently, if

we write

e = y - f (6#)

a

then

t

=

(l/n)E'E .

We illustrate with Example 1.

~l

a

=

1, 2, ... , M

EXAMPLE 1 (continued).

Fitting

(36)

Method. SAS Statements:

FRoe NLIN DATA=EXAMPLEl METHOD=GAUSS ITER=50 CONVERGENCE=1.E-13i PARMS B11=0 B12=0 B13=0 B31=0 B32=0 B33=0 Al=-9; A3=-li

P£AK=Al+Bll-X1+B12-X2+313-X3i BASE=A3+B31-Xl+B3Z-X2+B33-X3i MODEL Y1=LOG(PEAK/BASE)i

DER.Al =1/PEAKi

DER.Bll=l/PEAK-Xli DER.B31=-1/BASE*Xl i DER.B12=1/PEAK-X2i DER.B32=-1/BASE-XZi DER.B13=1/PEAK*X3i D£R.B33=-1/BASE*X3i OUTPUT OUT=VORK02 RESIDUAL=£li

Output:

S TAT 1ST I CAL A N A L Y S I S

S Y S T E M

NON-LINEAR LEAST SQUARES ITERATIVE PHASE

O.OOOOOOE+OO 0.000000£+00 ITERATION

o

DEPENDENT VARIABLE: Yl Bll B31 Al 0.000000£+00 0.000000£+00 -9.00000000 B12 832 METHOD: GAUSS-NEWTON B13 B33 O.OOOOOOE+OO O.OOOOOOE+OO RESIDUAL SS 72.21326991 16 -0.8386Z780 0.46865734 -1.98254583 -1.44241315

-0.19468166 -0.382996262.01535561 36.50071896

NON-LINEAR LEAST SQUARES SUMMARY STATISTICS NOT£: CONVERGENCE CRITERION MET.

S TAT 1ST I CAL

SOURCE REGRESSION RESIDUAL UNCORRECTED TOTAL (CORRECTED TOTAL) OF 7 217 224 Z23

A N A L Y S I S

SUM OF SQUARES 1019.72335676 36.50071896 1056.22407572 70.01946051

S Y S T £ M

DEPENDENT VARIABLE Yl MEAN SQUARE 145.61476525 0.16820608 3 PARAMETER Bl1 312 B13 D31 B32 333 Al ESTIMATE -0.83862780 -1.44241315 2.01535561 0.46865734 -0.19468166 -0.38299626 -1.98254583 ASYMPTOTIC STD. ERROR 1.37155782 1.87671707 1.44501283 0.12655505 0.21864114 0.09376286 1.03138455

(37)

"'#

e

₁

=

-1·98254583

-0.83862780

-1.44241315

2.01535561

0.46865734

-0.19468166

-0·38299626

6-2-7

and from Figure lb that

"'#

_e

₌

2 =

-1.11401781

0.41684196

-1·30951752

0.73956410

0.24777391

0.07675306

-0·39514717

Some aspects of these

co~putations

deserve comment.

In this instance,

the convergence of the modified Gauss-Newton method is fairly robust to the

choice of starting values so we have taken the simple expedient of starting with

8 =

08 +

[F'(08 ) F (08

)]-1

F'(08

)[y -

f

(08 )]

1

Ct'

is such that

is negative for some of the x

(38)

rigure lb. Second Equation of Example 1 Fitted by the Modified Gauss-Newton Method.

SAS Statements:

PROC NLIN DATA=EXAMPLEl METHOD=GAUSS ITER=50 CONVERGENCE=1.E-13i PARMS B21=0 B22=0 B23=0 831=0 B32=0 B33=0 A2=-3; A3=-1i

INTER=A2+B21*X1+B22*X2+B23*X3i BASE=A3+B31*X1+B32*X2+B33*X3i MODEL YZ=LOG(INTER/BASE) i

DER . A2 =1 lINTER ;

DER.BZ1=1/INTER*X1i DER.B31=-1/BASE*X1i DER.B22=1/INTER*X2; DER.B32=-1/BASE*X2; DER.B23=1/INTER*X3i DER.B33=-1/BASE*X3i OUTPUT OUT=VORK03 RESIDUAL=E2i

Output:

S TAT 1ST I CAL A N A L Y SIS S Y S T E M 4

NON-LINEAR LEAST SQUARES ITERATIVE PHASE

O.OOOOOOE+OO 0.000000£+00 ITERATION

o

DEPENDENT VARIABLE: Y2 B21 931 A2 0.000000£+00 O.OOOOOOE+OO -3.00000000 B22 B32 METHOD: CAUSS-NEVTON B23 B33 O.OOOOOOE+OO O.OOOOOOE+OO RESIDUAL SS 37.16988980 16 0.41684196 0.24777391 -1.11401781 -1.30951752

0.07675306 -0.395147170.73956410 19.70439405

NON-LINEAR LE~ST SQUARES SUMMARY STATISTICS NOTE: CONVERGENCE CRITERION MET.

S TAT 1ST I CAL

SOURCE REGRESSION RESIDUAL UNCORRECTED TOTAL (CORRECTED TOTAL) Dr 7 217 224 223

A N A L Y S ! S

SUM

or

SQUARES 265.3686590Z

19.70439405

285.07305307

36.70369496

S Y S T E M

D~PENDENT VARIABLE Y2 MEAN SQUARE 37.90980843 0.09080366 6 PARAMETER B21 B22 B23 B31 B32 833 A2 ESTIMATE 0.41684196 -1.30951752 0.73956410 0.24777391 0.07675306 -0.39514717 -1.11401781 ASYMPTOTIC STD. ERROR 0.44396622 0.60897020 O.:54937638 0.13857700 0.18207332 0.08932410 0.34304923

(39)

6-2-9

figure le. Contemporaneous Variance-Covariance Matria of Example 1 Estimated from Single Equation Residuals.

SAS Stl.tements:

DATA WORK04j MERGE WORK02 WORK03i KEEP T El E2i PROe MATRIX FW=20i FETCH E DATA=WORK04(KEEP=El E2) i

SIGMA=E''l!EIf2Z4j PRINT SIGMAi P=HALF(INV(SIGMA»i PRINT Pi Output:

SIGMA

S TAT 1ST I CAL

COLl

A

N A

L

Y

SIS

COL2

S Y S T E M 7

ROWl ROW2

p ROWl ROW2

0.1&29496382006 0.09015433203941

eOLl 3.764814163903

o

0.09015433203941 0.08796604486025

(40)

1

8

Ct

::: 08

Ct +

0" [F'(08 ) F (08

Ct Ct Ct Ct Ct

)J-~'(Oe

Ct

)[y

Ct - fCt

(08 )]

Ct

is in range to avoid this difficulty.

Thus, this situation is not a problem

for properly written code.

Other than cluttering up the output (suppressed

0.09015433203941 )

0.08796604486025

another approach to this problem. Lastly, we compute

(

0.1629496382006

~:::

0.09015433203941

in the figures), the

SAS

code seems to behave reasonably well.

See Problem 7

for

as shown in Figure lc.

For later use we compute

(

3.764814163903

1>:::

0

- 3 .85846955764 )

3.371659857133

. .... -1 .... , ....

Wl

th!:

::: P P.

0 The set of

M

regressions can be arranged in a single regression

y :::

f(8°)

+

e

by writing

f

1 (e

l

)

fee)

f

2 (8

2 )

:::

nM

f~(eM)

1 e

2 e :::

(41)

8

1 8==

8

2 e

_.:-1:

1

P

with

P

==

~==l

PO"

In order to work out the variance-covariance matrix of e

let us review Kroneker product notation.

If

A

is an k by

~

matrix and

B

is m by n then their Kroneker product,

denoted as

A ® B

is the

kIn

by

In

matrix

A®B==

The operations of matrix transposition and Kroneker product formation commute;

viz.

(A

®

B)

I ==

(A

I ®

B') •

If A and C are conformable for multiplication, that is, C has as many rows

as A has columns, and Band D are conformable as well then

(A

®

B) (C

®

D)

==

(AC

®

BD) •

It follows immediately that if both A and B are square and invertable then

(42)

In this notation, the variance-covariance matrix of the errors is

C(e

l

, e{)

C(el,e~)

C(el'~)

C(e

2 , e' )

1

C(e2,e~)

C(e2'~)

C(e,e') =

.

C(~,

e' )

_C(eM,e~) _C(~,~)

1

0"11 I

0"12

I

O"lM I

0"21 I

0"22 I

02M I

=

DM1

I

=

t ~ I ;

the identity is n by n while

~

is M by M so the resultant

~

0 I is

riM

by

riM •

-1 -1

Factor

t

as

t

=

P'P

and consider the rotated model

(P0I)'y= (P0I)'f(e)

+

(P0I)'e

or

"y"

=

"f"

(e)

+

"e"

Since

C("e", "e"') =

(P@I)'(~@I)(P@I)

= L_ 0 I M~ n n

(43)

6-2-13

is simply a univariate nonlinear model and 6° can be estimated by minimizing

S(6,l:)

=

["y" - "f"(6)J'["y" - "f"(e)J

=

[y

f(e)J'(p 0 I)'(P 0 I)[y - f(6)J

=

[y

f(e)]'(E-

1 0 I)[y - f(8))

Of course l: is unknown so one adopts the obvious expedient (Problem

4)

of

replacing l: by

!:

and estimating

eo

by

e

minimizing

S(6,~)

.

These ideas are easier to

L~plement

if we adopt the multivariate notational

scheme rather than the "seemingly unrelated" regressions scheme.

Accordingly,

let

t = 1, 2, ... , n

(44)

8 =

whence the model may be written as the multivariate nonlinear regression

t = 1, 2, ... , n .

In this scheme,

To see that this is so, let

cr~e

denote the elements of

~-l

and write

=

~=l ~l ~a=l cra@[Y~t

-

f~(Xt,e~))

[Yet - fa(Xt,ee)]

=

~l r~=l d:i'~[y~

-

f(e~)]/[YIa

- f(8

_e

)J

= [y -

f(e)]/(~-l

®

I)[y - f(e)]

The advantage of the multivariate notational scheme in writing code

derives fran the fact that it is natural to group observations (Yt,x

t

) on the

same subject together and process them serially for t = 1, 2, ... , n.

With

s(e,~)

written as

one can see at sight that it suffices to fetch (Yt'x

_t

),

compute

[Y

(45)

6-2-15

The notation is also suggestive of a transformation that permits the

use of univariate nonlinear regression programs for multivariate computations.

-1 -1

Observe that if L:

factors as

1: = pIp

then

Writing

p(~)

to denote the

~-th

row of P we have

S(s,r;)

=

I:~l ~l[P(~)Yt

-

P(~)f(xt,S)f

One now has S(S,L:) expressed as the sum of squares of univariate entities,

what remains is to find a notational scheme to remove the double summation.

To this end, put

s

=

M(

t-l)

+ ~

"X II

=

S

for

~

=

1, 2, ... , M

and t

=

1, 2, ... ,

n whence

( S

)

~nM [lly II _ IIf II

("x

II,

S)

J2

S

,L:

=

'"'s=l

s

We illustrate these ideas with the example.

EXAMPLE 1 (continued).

Recall that the model is

(46)

f Q' (x , 8,)

=

.tn( (aQ'

+

b (Q' ) )/

(-1

+

b ( 3 )x )]

Q'

=

1, 2

8'

=

(a

_l

, b

ll

, b

12 , b

13 , b

31 , b

32 , b

33 )

1 e'

=

(a

2 , b

21 , b

22 , b

23 , b

31 , b

32 , b

33 )

2 As the model is written, the notation suggests that b(3) is the same for both

Q'

=

1 and Q'

=

2 which up to now has not been the case.

To have a notation

that reflects this fact write

fQ'(x,8Q')

=

.tn((aQ'

+

b(Q'))/(-l

+

b~(3)x)J

Q'

=

1, 2

8'

=

(a

l

, b

i l,

b

12 , b

13 , b

131 , b

132 , b

133 )

1 8'

=

(a

2 , b

21 , b

22 , b

23 , b

231 , b

232 , b

233 )

2 to emphasize the fact that the equality constraint is not imposed.

The

multivariate model is, then,

(47)

e

=

and

Y

t

=(Y

lt\

'

Y

_{2t )}

for t

=

1 we have

a

l b l l b

12 b

13

b

131

b

132

b

133

a 2 b

21

b 22 b

23

b

231

b

232

b

233

e_t

=

,

x

_t

as before.

To illustrate,

fro~

Table la

Y

=

(.tn(0.662888/0.056731 ))

=

t

tn(0.280382/0.056731)

and for t

=

2 (

2.4582 9)

1.59783

_

(.tn(0.644427/0.103444))

=

(1.82933 )

Y

t -

.tn(0.252128/0.013444)

0.89091

as previously from Tables la and lb we have

(

2.11747)

xl

=

1.80731

0.81476

(

1.5

8

99

0 )

,

x

₂

=

1.27974

0.28719

To illustrate the scheme for minimizing

s(e,~)

using a univariate nonlinear

program, recall that

A

(3.7658

p -- 0

-3.8585 )

(48)

whence

"y

II

=

(3.7648, -3.8585)

(2.45829)

=

_3·08980

1 1.59783

"y

II

=

(

0,

3.3716)

(2.45829)

=

_5·38733

2 1·59783

"y

II

=

(3.7648, -3.8585)

C·82933)

=

_3·44956

3 0.89091

II " (

0,

3.3716)

(1..829

33)

3·00382

Y4

=

0.89091

"x "

₁

=

U. 7648, -3·8585, 2.11747, 1.80731, 0.81476) ,

"

x

"

(

0,

3.3716, 2.11747, 1.80731, 0.81476) ,

2 =

"x "

=

(3.7648, -3.8585, 1.58990, 1.27974, 0.28719)'

3

II "

0,

3.3716, 1.58990, 1.27974, 0.28719)'

x4

=

"f"("xl",e)

₌

(3.7648)

.tn[

_(a1

+ X

1 b(1))/(-1

+

x{b

1U

))J

-U·8585) .tn[(a2

+

x

₁

b(2))/(-1

+

x{b

2U

))]

"f"("x

₂

",e)

=

(3.3716) .tn[(a

₂

+

x{b(2))/(-1

+

x

₁

b

2U

))]

SAS code to implement this scheme is shown in Figure 2a together with the

resulting output.

Least squares methods lean rather heavily on normality for their validity.

Accordingly, it is a sensible precaution to check residuals for evidence of

severe departures from normality.

Figure 2a includes a residual analysis of

the unconstrained fit.

There does not appear to be a gross departure from

(49)

6-2-19

Figure 2a. Example 1 Fitted by Multivariate Least Squares, Unconstrained.

SAS Statements:

DATA ~ORKOli SET EXAMPLE1i

P1=3.764814163903i P2=-3.8~846955764i Y=Pl~Yl+P2~Y2i OUTPUTi Pl=Oi P2=3.371649857133i Y=P1-Yl+P2-Y2i OUTPUTi DELETEi PROC NLIN DATA=WORKOI METHOD=GAUSS ITER=50 CONVERGENCE=I.E-8i PARMS 311=-.8 BI2=-1.4 B13=2 8131=.5 BI32=-.2 8133=-.4

B21=.4 822=-1.3 823=.7 8231=.2 B232=.1 B233=-.4 Al=-2 A2=-li A3=-li

PEAK =Al+BII-X1+BI2*X2+813-X3i BASE1=A3+BI31*Xl+B132~X2+BI33~X3i INTER=A2+821-Xl+B2Z*X2+B23*X3i BASEZ=A3+B231-Xl+B23Z*X2+B233-X3i MODEL Y=Pl*LOG(PEAK/BASE1)+P2~LOG(INTER/BASE2);

DER.Al =PI/PEAKi DER.AZ =PZ/INTER; DER.B11=PI/PEAK*X1i DER.B21=PZ/INTER*Xli DER.B1Z=PI/PEAK*XZi DER.BZZ=P2/INTER-XZi DER.BI3=PI/PEAK-X3i DER.BZ3=P2/INTER*X3i DER.B131=-P1/8ASEI-Xli DER.8231=-P2/BASEZ-Xl; DER.8132=-PI/BASEI-XZi DER.B232=-P2/BASE2-X2i DER.8133=-PI/BASEI-X3i DER.B233=-P2/BASE2*X3; OUTPUT OUT=WORK02 RESIDUAL=EHATi

PRCC UNIVARIATE DATA=WORK02 PLOT NORMALi VAR EHATi 10 Ti

Output:

S T A TIS TIC A L A N A L Y S I S S Y S T E M NON-LINEAR LEAST SQUARES ITERATIVE PHASE

DEPENDENT VARIABLE: Y METHOD: GAUSS-NEVTON

ITERATION 811 BIZ 813 RESIDUAL 55

8131 8132 8133

821 B2Z B23

8231 8232 8233

At A2 1

o

6 -0.80000000 0.50000000 0.40000000 0.20000000 -2.00000000 -2.98669756 0.26718356 0.20848925 0.18931302 -1.52573841 -1.40000000 -0.20000000 -1.30000000 0.10000000 -1.00000000 0.90158533 0.07113302 -1.33081849 0.10756268 -0.96432128 2.00000000 -0.40000000 0.70000000 -0.40000000 1.66353998 -0.4101324Z 0.8~048354 -0.40539911 631.16222217 442.65919896

(50)

S TAT 1ST 1 CAL A N A L Y S I S S Y S T E M 2

NON-LINEAR LEAST SQUARES SUMMARY STATISTICS DEPENDENT VARIABLE Y SOURCr: REGRESSION nr:SIDUAL UNCORRECTED TOTAL (CORRECTED TOTAL) DF 14 434 448 447

SUM OF SQUARES 6540.63880955 442.65919896 6983. H800851 871.79801949 MEAN SQUARE 467.18848640 1.01995207

PARAMETER ESTIMATE ASYMPTOTIC STD. ERROR